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FullSimplify question

• To: mathgroup at smc.vnet.net
• Subject: [mg73619] FullSimplify question
• From: "dimitris" <dimmechan at yahoo.com>
• Date: Fri, 23 Feb 2007 04:34:23 -0500 (EST)

Consider the expression Tan[x] - Cot[x]

Then

Simplify[Tan[x] - Cot[x]]
Tan[x]-Cot[x]

Simplify always tries to minimize the value returned by the setting of
the ComplexityFunction. This means that the functions Tan and Cot will
not be present in the following result.

So

Simplify[Tan[x] - Cot[x], ComplexityFunction ->
             (Count[#, _Tan | _Cot, Infinity]&)]
-Cos[2 x] Csc[x] Sec[x]

When we explicitly INCLUDE TrigToExp in the list of transformations to
be applied, we get a result that does not contain any trig functions.

Simplify[Tan[x] - Cot[x], ComplexityFunction ->
   (Count[#, _Tan | _Cot | _Sin | _Cos | _Csc | _Sec, Infinity]&),
         TransformationFunctions -> {TrigToExp}]
(I*(E^((-I)*x) - E^(I*x)))/(E^((-I)*x) + E^(I*x)) + (I*(E^((-I)*x) +
E^(I*x)))/(E^((-I)*x) - E^(I*x))

Now consider the expression TrigToExp[ArcSinh[2]/ArcCsch[2]] from a
recent post of David
Cantrell.

Using Chris Chiasson's code from another recent post we get

Developer`ClearCache []

 Block[{$Output = {OpenWrite["C:\\msgStream.m"]}},
 TracePrint[FullSimplify[TrigToExp[ArcSinh[2]/ArcCsch[2]]],
 TraceInternal -> True]; Close /@ $Output];
 Thread[Union[Cases[ReadList["C:\\msgStream.m",
 HoldComplete[Expression]],
symb_Symbol /; AtomQ[Unevaluated[symb]] &&
 Context[Unevaluated[symb]] === "System`" :> HoldComplete[symb], {0,
Infinity},
 Heads -> True]], HoldComplete]

HoldComplete[{Abs, AiryAi,
   AiryAiPrime, AiryBi, AiryBiPrime, Algebraics, Alternatives, And,
Apart, \
Apply, ArcCos, ArcCosh, ArcCot, ArcCoth, ArcCsc, ArcCsch, ArcSec,
ArcSech,
    ArcSin, ArcSinh, ArcTan, ArcTanh, Arg, Assumptions, AtomQ,
Attributes,
    Automatic, BesselI, BesselJ, BesselK, BesselY, Beta,
BetaRegularized,
     Binomial, Blank, BlankNullSequence, Block, Ceiling, ChebyshevT,
    ChebyshevU, Coefficient, CoefficientList, Complement, Complex, \
ComplexityFunction, CompoundExpression, Condition, Conjugate, Cos,
    Cosh, CoshIntegral, CosIntegral, Cot, Coth, Cross, Csc,
    Csch, =C2=B0, Denominator, Derivative, DiracDelta, Dispatch, Divide,
    DivisorSigma, \[ExponentialE], EllipticE, EllipticF, EllipticK, \
EllipticNomeQ, EllipticPi, Equal, Erf,
    Erfc, Erfi, EulerE, EulerPhi, EvenQ, ExcludedForms, Exp, Expand, \
ExpandAll,
    ExpIntegralE, ExpIntegralEi, Exponent, ExpToTrig, Extension,
Factor, \
FactorComplete, Factorial, Factorial2, FactorInteger,
FactorSquareFree, \
FactorTerms, Fail, False, Fibonacci, First, Floor,
    FractionalPart, FreeQ, FresnelC, FresnelS,
     FullSimplify, Function, Gamma, GammaRegularized,
GaussianIntegers,
    GCD, GegenbauerC, General, GoldenRatio, Greater,
    HarmonicNumber, Head, Heads, HermiteH, HoldAll,
     HoldAllComplete, HoldComplete, HoldFirst,
    HoldForm, HoldPattern, HoldRest, Hypergeometric0F1, \
Hypergeometric0F1Regularized, Hypergeometric1F1,
    Hypergeometric1F1Regularized, Hypergeometric2F1, \
Hypergeometric2F1Regularized, HypergeometricPFQ, \
HypergeometricPFQRegularized, HypergeometricU, \[ImaginaryI],
Identity,
    If, Im, Inequality, =E2=88=9E, Integer, IntegerPart, IntegerQ,
    Integrate, Intersection, InverseEllipticNomeQ, InverseErf,
JacobiCD, \
JacobiCS, JacobiDC, JacobiDS,
    JacobiNC, JacobiND, JacobiNS, JacobiP, JacobiSC, JacobiSD,
    JacobiSymbol, JacobiZeta, LaguerreL, Last, LCM, LeafCount,
LegendreP, \
LegendreQ, Length, Less, LessEqual, List, Listable, Log, LogGamma,
    LogIntegral, Map, MatchQ, Max, MeijerG, MemberQ, Message,
MessageName,
    Mod, Module, Modulus, MoebiusMu, Multinomial, N, Negative, None,
    Not, Null, NumberQ,
    Numerator, NumericFunction, NumericQ, Off, On, OneIdentity,
Optional,
    Options, Or, Part, PartitionsP, Pattern, PatternTest, =CF=80, Plus,
Pochhammer,
     PolyGamma, PolyLog, PolynomialQ, Position, Positive,
    Power, PowerExpand, Prime, PrimePi, ProductLog, Protected,
Quotient,
    Rational, Rationalize, Re, ReplaceAll, Return,
RiemannSiegelTheta,
    RiemannSiegelZ, Root, Round, Rule, RuleCondition, RuleDelayed,
SameQ, \
SameTest, Sec, Sech, Select, Set, Sign, Sin, Sinh, SinhIntegral,
    SinIntegral, Slot, SphericalHarmonicY, Sqrt, StruveH, StruveL,
    Sum, Switch, Symbol, Tan, Tanh, TimeConstraint, Times,
    Together, TransformationFunctions, Trig, TrigReduce, True,
    TrueQ, Unevaluated, Union, UnitStep, UnsameQ, Variables,
    With, Zeta, $Assumptions, $ConditionHold, $MessageList, $Off}]



During the process of simplification the function TrigReduce is
called.
How is it possible to PREVENT the calling of TrigReduce?

Thanks a lot.

Dimitris

• Follow-Ups:
  • Re: FullSimplify question
    • From: "Chris Chiasson" <chris@chiasson.name>